Chapter 6 - Derivatives
Section 6.1 - The Derivative
Definition. Let \(I \subseteq \mathbb{R}\) be an interval \(f: I \rightarrow \mathbb{R}\) a function, and \(c \in I\). Then, the derivative of \(f\) at \(c\) is
provided that the limit exists. Thus, we obtain \(f'(x)\), or the derivative of \(f\), with a domain of all points \(c \in I\) where the limit exists.
Theorem. If \(f: I \rightarrow \mathbb{R}\) is differentiable at point \(c\), it is continuous at point \(c\).
Theorem. Let \(f, g: I \rightarrow \mathbb{R}\) be differentiable at \(c \in I\). Then,
- \((f+g)'(c) = f'(c) + g'(c)\)
- \((fg)'(c) = f'(c)g(c) + f(c)g'(c)\)
- \((\frac{f}{g})'(c) = \frac{f'(c)g(c) - f(c)g'(c)}{(g(c))^2}\)
Theorem. Let \(f: I \rightarrow \mathbb{R}\) and \(g: J \rightarrow \mathbb{R}\), with \(f(I) \subseteq J\). Then, if \(f\) is differentiable at \(c \in I\) and \(g\) is differentiable att \(f(c) \in J\), then \(f \circ g\) is differentiable at \(c\), and
Theorem. Suppose \(f: I \rightarrow \mathbb{R}\) is a one-to-one function on some interval \(I\). Then, with \(J = f(I)\) and \(f^{-1}: J \rightarrow \mathbb{R}\), for all \(x \in I\),
Lemma. \(f\) is continuous on some interval \(I\) if and only if it is monotonic on said interval.
Theorem. Suppose \(f: I \rightarrow \mathbb{R}\) is a one-to-one function on some interval \(I\). Then, with \(J = f(I)\) and \(f^{-1}: J \rightarrow \mathbb{R}\), if \(f\) is continuous on \(I\) and \(I\) is an interval, then \(f(I)\) is an interval, and \(f^{-1}\) is continuous on \(J\).
Theorem. Suppose \(f: I \rightarrow \mathbb{R}\) is differentiable at some \(c \in I\). Then, with \(J = f(I)\) and \(f^{-1}: J \rightarrow \mathbb{R}\), if \(f\) is differentiable at \(c\) and \(f'(c) \neq 0\), then \(f^{-1}\) is differentiable at \(d = f(c)\), and
Definition. Let \(I \subseteq \mathbb{R}\) be an interval and let \(f: I \rightarrow \mathbb{R}\). Then, \(f\) has a relative maximum (or local maximum) at some point \(c \in I\) if there exists some \(\delta\)-neighborhood \(V_\delta(c)\) such that for all \(x \in V_\delta(C) \cup I\), then \(f(x) \leq f(c)\). Relative minima are defined similarly.
Theorem. Let \(I\) be an interval and \(f: I \rightarrow \mathbb{R}\). Then, if \(f\) has a relative extremum at an interior point \(c \in I\), and if \(f'(c)\) exists, then \(f'(c) = 0\).
Corollary. Suppose \(f: [a, b] \rightarrow \mathbb{R}\) and \(g: [a, b] \rightarrow \mathbb{R}\) are both continuous on \([a, b]\) and differentiable on \((a, b)\) with \(a \neq b\). Then,
- Rolle's Theorem. If \(f(a) = f(b)\), then there exists at least one point \(c \in (a, b)\) with \(f'(c) = 0\).
- If \(f(a) = g(a)\) and \(f(b) = g(b)\), then there exists aat least one point \(c \in (a, b)\) such that \(f'(c) = g'(c)\).
Theorem. Mean Value Theorem. Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b)\), with \(a \neq b\). Then, there exists at least one \(c \in (a, b)\) with
Theorem. Suppose \(f: I \rightarrow \mathbb{R}\) is differentiable on I. Then,
- If \(f'(x) > 0\) for all $x \in $I, then \(f\) is strictly increasing on \(I\).
- If \(f'(x) = 0\) for all $x \in $I, then \(f\) is constant on \(I\).
- If \(f'(x) <> 0\) for all $x \in $I, then \(f\) is strictly decreasing on \(I\).
Theorem. Cauchy Mean Value Theorem. Let \(f, g\) be continuous on \([a, b]\) and differentiable on \((a, b)\). If \(g'(x) \neq 0\) for all \(x \in (a, b)\), and \(a \neq b\), then there exists at least one point \(c \in (a, b)\) such that